The energy required is the same in theory but not in practice.
Centrifugal compressors are delivering two things at the same time. Volume and pressure. A single stage is limited as to the pressure it can deliver and no amount of extra power (or size of the compressor if it is the same design) will really increase the pressure.
Centrifugal compressors are NOT positive displacement like a piston pump. They can 'stall', they can 'choke' and since the only thing that keeps the working fluid (air) moving through the compressor (pump) is momentum once they hit the limit of the design any excess 'flow' simply bleeds back around the edges or walls of the passages.
This is actually a gradual process as the limit is approached and shows up as the rapidly decreasing efficiency of the compressor as it's limits are reached.
This also brings in the advantage of the two stage compressor. Two compressors acting in series, each operating at a relatively high efficiency will need less power to compress the same amount of air to the same pressure as a single compressor operating at a low level of efficiency.
Since the difference in power between the actual work done in compressing the air and the power needed by the compressor shaft mostly shows up as extra heat in the charge air, using two stages also means a lower intake charge temperature compared to a single stage supercharger even if you could get one that would give you the pressure desired.
For most of WW II no single stage centrifugal compressor, even in the laboratory, would give the required pressure for high speed flight in the 20,000ft and up range.
Inter and after coolers are necessary to get anywhere the full benefit from high pressure ratios as Colin has said.
